Questions on the usage and meaning of words in mathematics, the names for mathematical entities, and other such questions.

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6
votes
1answer
66 views

When vectors act on scalars.

Background. I've been struggling through an introduction to differential geometry this semester. Recently, a tiny part of what we've been learning "clicked" for me, and to solidify this, I'd like ...
2
votes
1answer
36 views

Is a Linear Transformation a Vector Space Homomorphism?

I see the terms linear transformation and (vector space) homomorphism used more or less interchangeably, and the set (space) of linear transformations from V to W referred to as Hom(V, W) or ...
1
vote
0answers
11 views

What is the meaning of “mass defect” in measure theory?

What does the term "mass defect" mean in measure theory? I stumbled upon it in the context of weak convergence and the dominated convergence theorem, but I haven't seen it defined anywhere.
1
vote
1answer
93 views

Is there a well-known type of differential equation consisting of $y$, $y'$, and $y''$ multiplied together? [closed]

Is there some sort of well known type of differential equation consisting of first and second derivatives multiplied together? For example (I just made this up): $$y''(y')^2-y-x^2=0$$ Edit: Are ...
1
vote
1answer
46 views

Some weaker axiom than “no nontrivial zero divisors.”

I would like to know if there a standard term for or well-known applications of the following axiom for rings or semigroups with zero (which is weaker than the "no nontrivial zero divisors" axiom): ...
3
votes
1answer
44 views

Is there a special name for matrices $A^T A$ and $A A^T$?

I'm looking for a special name for matrices $A^T A$ and $A A^T$. "Symmetric" and "Positive semi-definite" are too general terms. These matrices have special properties, so they should have a special ...
1
vote
3answers
46 views

how to call a and b when a+b=1?

I guess it's a simple question, but it really escaped my memory. If $a + b =1$, then how can I call those $a$ and $b$ numbers? $a$ is not an inversion of $b$, and it's not reciprocal of $b$.. but ...
2
votes
1answer
40 views

Well-posed vs Well-conditioned

What's the difference between a well-posed (ill-posed) and well-conditioned (ill-conditioned) problem ?` Here is my finding up to now: "Even if a problem is well-posed, it may still be ...
1
vote
0answers
48 views

Mathematical structures with name reffering to a country

I am looking for a list of mathematical structures (not theorems) that refer to a country or nationality. I only know of Polish spaces and Polish groups. Does anyone have other examples? Note: many ...
1
vote
1answer
46 views

Why did mathematicians name a functional that assigns number to function as a “distribution”?

Why did people name it as a "distribution"? I don't see the reason. My instructor told us don't bother with this strange name, but I guess maybe I will have a better understanding if I know the ...
4
votes
0answers
17 views

Terminology: order vs. degree (in general)

The word degree comes from Latin degradus (through French), which means something like step down. The word order comes from Latin ...
5
votes
5answers
513 views

Is there an intuitive, not-too-mathematical way of thinking about limit points? [duplicate]

so I know this question has been asked sooo many times. But I just have a few questions in particular, which despite searching, I haven't found an answer to. I appreciate any help. Book's definition: ...
6
votes
4answers
95 views

Is the logarithm of $\aleph_0$ infinite?

In classical mathematics $2^{\aleph_0}=\aleph_1$, right? So if $2^x=\aleph_0$, what does $x$ equal? In other words, can we define a logarithm for $\aleph_0$, and what should it be. Is it infinite? ...
0
votes
2answers
20 views

Comparing Open Bases and Covers

In Topology, I see a resemblance and similarity between open bases and open covers. Although this is a short question, what is the defining difference between the two that sets them apart? ...
2
votes
0answers
31 views

Rings where action of automorphisms on maximal ideals is transitive

If $R$ is a commutative ring, $\alpha: R \to R$ an automorphism of $R$, and $M$ a maximal ideal of $R$, then $\alpha(M)$ is also a maximal ideal of $R$ with the same quotient field. So the group of ...
0
votes
2answers
23 views

Does a one-to-one function exhibit “injectiveness” or “injectivity”?

I'm preparing some tutorials for students and I'm faced with writer's block. If I want to say a function is injective/one-to-one, would the function demonstrate "injectivity" or "injectiveness"? ...
2
votes
0answers
39 views

What's the word for a number which is used to scale down a value?

I'm a programmer and I'm creating an API in which there is a parameter the user can pass in which scales down a value. So for example: ...
2
votes
0answers
66 views

Mathematical definition of the Hamiltonian function.

I'm reading this nice text on Calculus of Variations, by Peter Olver. In page $8$, he calls $$J[u] = \int_a^b L(x,u,u')\,{\rm d}x$$ the objective functional, and the integrand $L(x,u,u')$ the ...
1
vote
0answers
22 views

Does the differential operator in the heat equation have a name?

Does the operator $$\frac {\partial}{\partial t} - k\nabla^2$$ have a name?
3
votes
1answer
45 views

Is there a name for this curve? Or, how should I describe the behavior of this graph (in words)?

I simulated some results that look like this: but I don't want to include the plot (my advisor is keeping me to a strict limit on figures and these are minor intermediate results). Is there a name ...
1
vote
1answer
46 views

“Quotient” as a verb

People here do use "quotient" as a verb: I searched for "quotienting" and got 12,890 results. [Edit: It's not as bad as I thought. Apparently I didn't understand how the search function works. When I ...
1
vote
1answer
63 views

Where can I learn to define mathematical terms?

For example, take the following: The radian measure of a central angle of a circle is defined as the ratio of the length of the arc the angle subtends, s, divided by the radius of the circle, ...
-1
votes
1answer
60 views

Why are nodes and nodal sets called this way?

Nodes of standing waves are points where they are zero. Generally, nodal sets of Laplacian eigenfunctions are the sets of points where they are zero. Why is this the name for them (that is, why is ...
1
vote
0answers
30 views

A name for a particular covering map?

The quotient space of $\mathbb C$ obtained by identifying points differing by a Gaussian integer is topologically a torus. The map that takes each point in $\mathbb C$ to its corresponding point in ...
0
votes
1answer
37 views

Difference between “distribution” & “arrangement”.

Number of ways of Arrangement of $n$ different things into $r$ different groups is $$n!\binom{n - 1}{r - 1}$$. Number of ways of distribution of $n$ different things into $r$ different groups is the ...
2
votes
2answers
105 views

Use of either/or in maths

I have been using these two words for a long time, especially when representing the solutions to quadratic equations. But I am little confused. These terms are often used simultaneously, but it seems ...
2
votes
1answer
41 views

“Sharp” Inequalities

When we say that an inequality is sharp, does it mean that it is "the best" inequality we can get between the two quantities involved? For example, I read that we would say that the inequality $$ ...
0
votes
0answers
23 views

Notation: column/row projection function for matrix-like objects

If we have a $n$-tuple $\mathscr x$ $$\mathbf{x} := (x_i)_{i\in n}=(x_0,x_1,\ldots,x_{n-1})\in \prod_{i\in n}X_i$$ where $(X_i)_{i\in n}$ is an indexed family of sets and $x_i\in X_i$. We can ...
0
votes
0answers
95 views

Submodules $H$ satisfying: “if $ax \in H$ for some non-zero scalar $a$, then $x \in H$.”

Suppose $R$ is a commutative ring and that $X$ is an $R$-module. Question. Is there a term for those $R$-submodules $H$ of $X$ satisfying the following? For all $x \in X$, if $ax \in H$ ...
1
vote
0answers
20 views

Definition of a minimal set of automorphisms generating an orbit?

By definition, the orbit of a vertex v in a graph G is the set of all vertices f (v) such that f is an automorphism of G. I wonder whether there is a definition for a minimal set S of automorphisms ...
0
votes
2answers
21 views

Is there another terminology to designate this?

Let $R$ be a principal ideal domain. Let $M$ be a finitely generated $R$-module. Then there exists a free $R$-submodule $F$ of $M$ such that $M=Tor(M)\oplus F$ and the ranks of such $F$'s are the ...
6
votes
2answers
120 views

If $a=b$ then $a+c=b+c$? [duplicate]

A friend of mine just asked me how to prove that if $a=b$ then $a+c=b+c$, where $a,b$ and $c$ are real numbers, I'm not sure what I should answer. I have a book called introduction to logic and to the ...
5
votes
2answers
187 views

Why is $\sinh$ often pronounced “shine”?

I talked to some guys from the UK and they told me that they would pronounce $\sinh$ as "shine". I am not a native english speaker so I don't know, but in my country we call this function "sintsh" ...
0
votes
1answer
26 views

Are there terminologies distinguishing these two ranks?

Definition 1. Let $R$ be a ring with invariant basis number and $M$ be a free $R$-module. Then, the rank of $M$ is the cardinality of an $R$-basis of $M$. ${}$ Definition 2. ...
3
votes
1answer
20 views

Order of a polynomial in $\mathbb F_q[x]$

I came across the term "order" in the context of $\mathbb F_q[x]$, specifically of irreducible polynomials. Does this mean order in the group theoretical sense? I tried to prove that every polynomial ...
2
votes
1answer
62 views

Positive and negative logical connectives

By inspecting the rules of inference for (intuitionistic) predicate calculus (or, alternatively, thinking about double negation translation), one sees that there is a certain dichotomy between two ...
2
votes
1answer
73 views

Name of function $(1+x)^n-1$

Is there any name for this formula $$(1+x)^n-1$$ When working with floating point numbers this can be calculated with much better precision for very small $|x|<1$ values using Taylor series ...
-3
votes
1answer
32 views

What is mutually disjoint sets

What is mutually disjoint sets? I know it has something to do with subsets but I don't know for sure.
4
votes
1answer
38 views

Terminology: Difference between Lemma, Theorem, Definition, Hypothesis, Postulate and a Proposition

Based on observation after reading few books and papers, I think that Lemma : Lemma contains some information that is commonly used to support a theorem. So, a Lemma introduces a Theorem and comes ...
2
votes
2answers
54 views

Why is it called the category of representations?

Let $A$ be a (Hopf) algebra. Let $C_A$ be a category whose objects are $A$-modules and whose morphisms are $A$-linear maps. This category is called "the category of representations". My question is: ...
0
votes
0answers
10 views

What is the nomenclature for the repeating part of a curve with n-repeating-peaks?

Below is a Google Trends search for "past papers", notice the curve has repeating portions where each repeat has three peaks at different levels. I want to know what the technical name of such a ...
3
votes
1answer
33 views

Convex functions up to reparametrization

I would like to know if there is a standard name for functions $f:[0,1]\to\mathbb R$ with the following convexity property: $$ \forall s<t<u\qquad f(t)\leq\max\{f(s),f(u)\}$$ (the fact that ...
0
votes
1answer
43 views

Lambert W-Function

Is there a standard name for the inverse of the Lambert W-Function, in the manner that the name "exponential function" is the name for the inverse function of the logarithmic function.
0
votes
0answers
35 views

Terminology for functions such that $f(x)\ge x$ for all $x$

Is there a common terminology for a real function $f$ such that $$f(x)\ge x$$ for all $x$? same question for the conditions $\forall x,f(x)>x$; $\forall x,f(x)\le x$; $\forall x,f(x)<x$. (I'm ...
3
votes
1answer
46 views

What does it mean for pullbacks to preserve monomorphisms?

If two arrows $f_A : A \to C$, $f_B : B \to C$ are monomorphisms, then their pullback arrows $p_A : P \to A$, $p_B : P \to B$ are monomorhisms too. Is that what is meant by pullbacks preserving ...
0
votes
1answer
28 views

How's this inertia called?

Let $E/F$ be an algebraic extension. Let $L_1,L_2$ be algebraically closed fields and $\sigma_1:F\rightarrow L_1,\sigma_2:F\rightarrow L_2$ be field monomorphisms. Define ...
2
votes
0answers
32 views

How is this commutator property $[a,b]=bx$ called?

If two elements $a,b$ commute like this $[a,b]=bx$ for some $x$ so that it can rewriten as $$ ba=a(b-x), $$ is there a term for such property? It is as if $a$ and $b$ almost commuted (but not quite) ...
1
vote
1answer
22 views

Mathematical Name for Physical Gauge Symmetries

In physics, when talking about a gauge transformation, we always mean two combined transformations. For example, a $U(1)$ gauge transformation is a combination of $$ \psi \rightarrow e^{ia(x)} \psi ...
1
vote
1answer
55 views

informal semantics regarding CH and AC

why is the assertion $\aleph_1=2^{\aleph_0}$ referred to as a hypothesis, whereas $$\forall \alpha( S_\alpha \ne \varnothing) \Rightarrow \prod_\alpha S_\alpha \ne \varnothing$$ is called an axiom? ...
0
votes
1answer
34 views

What are the names of these variations on the transpose of a matrix and symmetric matrices?

Is there a name for the operator that reflects a matrix over the diagonal running from the top-right to the bottom-left? For the moment, define this reflection of a matrix $A$ as $A^*$. Is there a ...